The Character Theory of a Complex Group

Abstract

We apply the ideas of derived algebraic geometry and topological field theory to the representation theory of reductive groups. Our focus is the Hecke category of Borel-equivariant D-modules on the flag variety of a complex reductive group G (equivalently, the category of Harish Chandra bimodules of trivial central character) and its monodromic variant. The Hecke category is a categorified analogue of the finite Hecke algebra, which is a finite-dimensional semi-simple symmetric Frobenius algebra. We establish parallel properties of the Hecke category, showing it is a two-dualizable Calabi-Yau monoidal category, so that in particular, its monoidal (Drinfeld) center and trace coincide. We calculate that they are identified through the Springer correspondence with Lusztig's unipotent character sheaves. It follows that Hecke module categories, such as categories of Lie algebra representations and Harish Chandra modules for G and its real forms, have characters which are themselves character sheaves. Furthermore, the Koszul duality for Hecke categories provides a Langlands duality for unipotent character sheaves. This can be viewed as part of a dimensionally reduced version of the geometric Langlands correspondence, or as S-duality for a maximally supersymmetric gauge theory in three dimensions.

Substantial revision based on referee comments, references updated, statements of main results unaffected

References in corpus (10)

• Higher Topos Theory
• The Omega Deformation, Branes, Integrability, and Liouville Theory
• The Classification of Two-Dimensional Extended Topological Field Theories
• D-branes and K-theory in 2D topological field theory
• Higher and derived stacks: a global overview
• Gauge Theory, Ramification, And The Geometric Langlands Program
• Noncommutative Counterparts of the Springer Resolution
• Geometric Langlands And The Equations Of Nahm And Bogomolny
• Loop Spaces and Langlands Parameters
• A survey of (\infty, 1)-categories

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• Symmetries of categorical representations and the quantum Ngô action
• Coherent Springer theory and the categorical Deligne-Langlands correspondence
• Beilinson-Bernstein localization over the Harish-Chandra center
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• A Derived Decomposition for Equivariant $D$-modules
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• Equivariant localization in factorization homology and applications in mathematical physics I: Foundations
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• Equivariant localization in factorization homology and applications in mathematical physics II: Gauge theory applications
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• Homological Mirror Symmetry for the universal centralizers I: The adjoint group case
• Affine Harish-Chandra bimodules and Steinberg--Whittaker localization